Why does the scalar field have the wrong sign mass term in its Lagrangian to spontaneously break? In a class, the professor said that there are "some" reasons why the scalar field should have had wrong sign mass term in its lagrangian.
What are those reasons?
(In undergraduate textbook, Griffiths only says 'God's computer returns an error message'. The professor also mentioned tachyonic mass which is imaginary. Is it relevant here?)
edit
I am dealing with following Lagrangian:
$\mathcal{L} = \frac{1}{2}[(\partial_\mu-\frac{iq}{\hbar c}A_\mu)\phi^*]
[(\partial_\mu+\frac{iq}{\hbar c}A_\mu)\phi] + \frac{1}{2}\mu^2 (\phi^*\phi)-\frac{1}{4}\lambda^2(\phi^*\phi)^2-\frac{1}{16\pi}F^{\mu\nu}F_{\mu\nu}
$
and the reason why I'm saying it has wrong sign mass is that it($+\frac{1}{2}\mu^2(\phi^*\phi$)) is opposite to the sign in usual scalar QED. 
(I edited Lagrangian because I accidently left out $\phi^4$term.)
 A: The wrong sign means that the minimum of the potential is not at $\phi = 0$. If the energy is bounded from below, you must have another term in the potential that has the right (positive) sign and that dominates at large values of $\phi$, for instance $\phi^4$. 
For simplicity, assume that you have a potential of the form $-m^2 \phi^2 + \lambda \phi ^4$. Then you can show that the minimum of the energy is obtained for a field $\phi$ that takes some constant value $\phi_0 \neq 0$. In the initial potential, you had a symmetry $\phi \rightarrow - \phi$ because only even powers are present. But this symmetry is lost in the vacuum where $\phi = \phi_0$. When there is a symmetry in an equation that is lost in the solutions of this equation, physicists say that there is spontaneous symmetry breaking. 
In summary, you see that the "wrong" sign in front of the mass term, that signals a local instability, has triggered a spontaneous breaking of some symmetry. 
If you know about the Higgs mechanism in the Standard Model, something similar happens. 
A: Spontaneously broken scalar theories have a $\phi^4$ term that requires a negative $\phi^2$ (mass) term in order to give the central hump of the characteristic Mexican hat potential. Without that hump, spontaneous symmetry breaking would not occur.
